Theorem isumclim3 12498

Description: The sequence of partial finite sums of a converging infinite series converge to the infinite sum of the series. Note that j must not occur in A. (Contributed by NM, 9-Jan-2006.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
isumclim3.1	|- Z = ( ZZ>= ` M )
isumclim3.2	|- ( ph -> M e. ZZ )
isumclim3.3	|- ( ph -> F e. dom ~~> )
isumclim3.4	|- ( ( ph /\ k e. Z ) -> A e. CC )
isumclim3.5	|- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )

Assertion

Ref	Expression
isumclim3	|- ( ph -> F ~~> sum_ k e. Z A )

Distinct variable groups:   A, j   j, k, M   ph, j, k   j, Z, k   j, F

Allowed substitution hints:   A( k)   F( k)

Proof of Theorem isumclim3

Dummy variables m x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isumclim3.3	. . 3 |- ( ph -> F e. dom ~~> )
2		climdm 12303	. . 3 |- ( F e. dom ~~> <-> F ~~> ( ~~> ` F ) )
3	1, 2	sylib 189	. 2 |- ( ph -> F ~~> ( ~~> ` F ) )
4		sumfc 12458	. . . 4 |- sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = sum_ k e. Z A
5		isumclim3.1	. . . . 5 |- Z = ( ZZ>= ` M )
6		isumclim3.2	. . . . 5 |- ( ph -> M e. ZZ )
7		eqidd 2405	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
8		isumclim3.4	. . . . . . 7 |- ( ( ph /\ k e. Z ) -> A e. CC )
9		eqid 2404	. . . . . . 7 |- ( k e. Z |-> A ) = ( k e. Z |-> A )
10	8, 9	fmptd 5852	. . . . . 6 |- ( ph -> ( k e. Z |-> A ) : Z --> CC )
11	10	ffvelrnda 5829	. . . . 5 |- ( ( ph /\ m e. Z ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
12	5, 6, 7, 11	isum 12468	. . . 4 |- ( ph -> sum_ m e. Z ( ( k e. Z |-> A ) ` m ) = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
13	4, 12	syl5eqr 2450	. . 3 |- ( ph -> sum_ k e. Z A = ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) )
14		seqex 11280	. . . . . . 7 |- seq M ( + , ( k e. Z |-> A ) ) e. _V
15	14	a1i 11	. . . . . 6 |- ( ph -> seq M ( + , ( k e. Z |-> A ) ) e. _V )
16		isumclim3.5	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> ( F ` j ) = sum_ k e. ( M ... j ) A )
17		fzssuz 11049	. . . . . . . . . . . . . 14 |- ( M ... j ) C_ ( ZZ>= ` M )
18	17, 5	sseqtr4i 3341	. . . . . . . . . . . . 13 |- ( M ... j ) C_ Z
19		resmpt 5150	. . . . . . . . . . . . 13 |- ( ( M ... j ) C_ Z -> ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A ) )
20	18, 19	ax-mp 8	. . . . . . . . . . . 12 |- ( ( k e. Z |-> A ) |` ( M ... j ) ) = ( k e. ( M ... j ) |-> A )
21	20	fveq1i 5688	. . . . . . . . . . 11 |- ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m )
22		fvres 5704	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> ( ( ( k e. Z |-> A ) |` ( M ... j ) ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
23	21, 22	syl5reqr 2451	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. ( M ... j ) |-> A ) ` m ) )
24	23	sumeq2i 12448	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m )
25		sumfc 12458	. . . . . . . . 9 |- sum_ m e. ( M ... j ) ( ( k e. ( M ... j ) |-> A ) ` m ) = sum_ k e. ( M ... j ) A
26	24, 25	eqtri 2424	. . . . . . . 8 |- sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = sum_ k e. ( M ... j ) A
27		eqidd 2405	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) = ( ( k e. Z |-> A ) ` m ) )
28		simpr 448	. . . . . . . . . 10 |- ( ( ph /\ j e. Z ) -> j e. Z )
29	28, 5	syl6eleq 2494	. . . . . . . . 9 |- ( ( ph /\ j e. Z ) -> j e. ( ZZ>= ` M ) )
30		simpl 444	. . . . . . . . . 10 |- ( ( ph /\ j e. Z ) -> ph )
31		elfzuz 11011	. . . . . . . . . . 11 |- ( m e. ( M ... j ) -> m e. ( ZZ>= ` M ) )
32	31, 5	syl6eleqr 2495	. . . . . . . . . 10 |- ( m e. ( M ... j ) -> m e. Z )
33	30, 32, 11	syl2an 464	. . . . . . . . 9 |- ( ( ( ph /\ j e. Z ) /\ m e. ( M ... j ) ) -> ( ( k e. Z |-> A ) ` m ) e. CC )
34	27, 29, 33	fsumser 12479	. . . . . . . 8 |- ( ( ph /\ j e. Z ) -> sum_ m e. ( M ... j ) ( ( k e. Z |-> A ) ` m ) = ( seq M ( + , ( k e. Z |-> A ) ) ` j ) )
35	26, 34	syl5eqr 2450	. . . . . . 7 |- ( ( ph /\ j e. Z ) -> sum_ k e. ( M ... j ) A = ( seq M ( + , ( k e. Z |-> A ) ) ` j ) )
36	16, 35	eqtr2d 2437	. . . . . 6 |- ( ( ph /\ j e. Z ) -> ( seq M ( + , ( k e. Z |-> A ) ) ` j ) = ( F ` j ) )
37	5, 15, 1, 6, 36	climeq 12316	. . . . 5 |- ( ph -> ( seq M ( + , ( k e. Z |-> A ) ) ~~> x <-> F ~~> x ) )
38	37	iotabidv 5398	. . . 4 |- ( ph -> ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x ) = ( iota x F ~~> x ) )
39		df-fv 5421	. . . 4 |- ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( iota x seq M ( + , ( k e. Z |-> A ) ) ~~> x )
40		df-fv 5421	. . . 4 |- ( ~~> ` F ) = ( iota x F ~~> x )
41	38, 39, 40	3eqtr4g 2461	. . 3 |- ( ph -> ( ~~> ` seq M ( + , ( k e. Z |-> A ) ) ) = ( ~~> ` F ) )
42	13, 41	eqtrd 2436	. 2 |- ( ph -> sum_ k e. Z A = ( ~~> ` F ) )
43	3, 42	breqtrrd 4198	1 |- ( ph -> F ~~> sum_ k e. Z A )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   _Vcvv 2916   C_ wss 3280   class class class wbr 4172   e. cmpt 4226   dom cdm 4837   |` cres 4839   iotacio 5375   ` cfv 5413  (class class class)co 6040   CCcc 8944   + caddc 8949   ZZcz 10238   ZZ>=cuz 10444   ...cfz 10999   seq cseq 11278   ~~> cli 12233   sum_csu 12434

This theorem is referenced by:  esumcvg  24429

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435